ARTICLES
PUBLISHED ONLINE: 24 MARCH 2015 | DOI: 10.1038/NCHEM.2204
Quantum dynamical resonances in low-energy
CO( j = 0) + He inelastic collisions
Astrid Bergeat1,2, Jolijn Onvlee3, Christian Naulin1,2, Ad van der Avoird3* and Michel Costes1,2*
In molecular collisions, long-lived complexes may be formed that correspond to quasi-bound states in the van der Waals
potential and give rise to peaks in the collision energy-dependent cross-sections. They are known as ‘resonances’ and
their experimental detection remains difficult because their signatures are extremely challenging to resolve. Here, we
show a complete characterization of quantum-dynamical resonances occurring in CO–He inelastic collisions with rotational
CO( j = 0 → 1) excitation. Crossed-beam scattering experiments were performed at collision energies as low as 4 cm−1,
equivalent to a temperature of 4 K. Resonance structures in the measured cross-sections were identified by comparison
with quantum-mechanical calculations. The excellent agreement found confirms that the potential energy surfaces
describing the CO–He van der Waals interaction are perfectly suitable for calculating state-to-state (de)excitation rate
coefficients at the very low temperatures needed in chemical modelling of the interstellar medium. We also computed
these rate coefficients.
I
nelastic collisions between molecules are fundamental processes in
which energy is transferred between their relative translational
motion and their internal degrees of freedom. The probability
that the molecules enter the collision in a given state and exit in
another state is expressed in terms of the state-to-state integral
cross-section (ICS).
The ICS is zero below the ‘threshold energy’—the energy needed
to excite the higher state—if the final state is higher in energy than
the initial one. Above this threshold, classical mechanics predicts
that the ICS rises sharply to a maximum and then decreases
smoothly at higher energies1. However, quantum mechanics
shows that at low kinetic energies (a few cm−1) the inelastic scattering cross-sections of simple molecular species of astrophysical
importance (such as CO, O2 , CN, OH, H2O and NH3) colliding
with H2 molecules or He atoms do not follow such a simple
threshold law, but are instead highly structured2–13. This is the
case, in particular, in the vicinity of the thresholds of the lowest molecular rotational excitations. Such behaviour of the ICS is important
when the collision energy is comparable to the depth of the shallow
van der Waals well, which characterizes the interaction potential
between the colliding partners. Furthermore, the ICSs become
very sensitive to the shape of this potential under such conditions.
A crucial parameter in the description of a collisional process is
the orbital angular momentum, L = μvrb, the product of the reduced
mass μ, the relative velocity vr and the impact parameter b.
Parameter b is the distance of closest approach of the molecules
in the absence of any interaction (b = 0 for head-on collisions). In
quantum mechanics, the modulus of the orbital angular momentum
is |L| = {l(l + 1)}1/2ħ, where l can only take integer values and ħ is the
reduced Planck constant. Each value of l is associated with a particular
quantum-scattering state, or ‘partial wave’. A consequence of this
rotational relative motion is that a fraction of the initial translational
energy goes into this motion, namely the centrifugal energy, and
is no longer available for the approach of the collision partners. This
is taken into account by defining an effective potential Veff that is
experienced by molecules during a collision of orbital angular
momentum l: Veff (R; l) = V(R) + ER , where V(R) is the van der
Waals potential and ER = l(l + 1)ħ 2/2μR 2 is the centrifugal energy.
Figure 1 presents a schematic view of what happens during CO–He
collisions. It shows that the addition of ER to the potential V(R) produces a so-called centrifugal barrier. This picture is somewhat simplified. In reality, the van der Waals potential V depends not only on the
distance R between the colliding partners, but also on the orientation
of the CO molecule and the CO bond length.
Truly bound states exist in the van der Waals well of the potential
for energies lower than the dissociation limit. When the energy is
above the dissociation limit—as it is during collisions—resonances
or quasi-bound states may occur. If, for instance, the CO molecule
enters the collision in its rotational ground state with quantum
number j = 0 with a collision energy higher than the energy of its
j = 1 state but lower than the centrifugal barrier, a so-called ‘shape’
(or ‘orbiting’) resonance may occur for specific energies. When that
happens, the system quantum-mechanically tunnels through the centrifugal barrier, stays trapped in a quasi-bound state for a certain time,
and finally exits again by tunnelling, either in its j = 0 state (an elastic
collision) or in its j = 1 state (an inelastic collision). Such a shape resonance, in which the colliding partners interact for a relatively long
time, leads to an increase in the probability that the CO molecule
becomes excited to its j = 1 state, that is, to a peak in the ICS at the
energy where the resonance occurs.
Another type of resonance may also occur at different collision
energies. When, for example, the collision energy lies above the
CO( j = 1) limit but below the dissociation limit corresponding to
j = 2, the complex can still be temporarily trapped in the j = 2
state. Energetically, this is allowed at small distances where the
attractive potential is negative and more kinetic energy is available.
It cannot decay in this CO( j = 2) state, however, because this is energetically not allowed at a long distance. It can decay to j = 1, again
with an increased probability due to the longer lifetime of the
quasi-bound complex. This is known as a Feshbach resonance14–16.
The resonance behaviour influences the magnitude of the rate
coefficients for state-to-state (de)excitation, especially at very low
1
Université de Bordeaux, Institut des Sciences Moléculaires, Talence Cedex 33405, France. 2 CNRS, UMR 5255, Talence Cedex 33405, France.
Radboud University Nijmegen, Institute for Molecules and Materials, Heijendaalseweg 135, Nijmegen 6525 AJ, The Netherlands.
* e-mail: [email protected]; [email protected]
3
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349
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Effective potential (cm–1)
V2
10
(II)
(I)
0
V1
j = 2, I = 3
V0
j = 1, I = 5
j = 0, I = 0
–10
–20
1
2
3
R/Re
Figure 1 | Effective potentials illustrating shape and Feshbach resonances.
The effective potentials for CO–He collisions are schematically represented
for j = 0, l = 0 (V0), j = 1, l = 5 (V1) and j = 2, l = 3 (V2). Re is the position of
the minimum of the well and the horizontal lines at long R/Re distances are
the respective dissociation limits. Quasi-bound states corresponding to the
dominant contributions in the wavefunctions displayed in Fig. 4 are also
represented. Resonances occur when the incident energy coincides with
such a state. Two different situations can occur. For the V1 curve, the quasibound state I is below the centrifugal barrier but above the dissociation limit:
in this case, the complex can access this state by tunnelling through the
centrifugal barrier, and then decays to j = 1, resulting in rotational excitation
of CO. This is known as a ‘shape’ (or ‘orbiting’) resonance. For the V2 curve,
state II is below the dissociation limit; the complex can ephemerally access
this state, which is energetically allowed at small distances. It cannot
dissociate in its j = 2 state because this is not energetically allowed. It can,
however, decay to j = 1; this is known as a Feshbach resonance.
1
1
ET = μvr2 = μ(v12 + v22 − 2v1 v2 cosχ)
2
2
(1)
Consideration of equation (1) shows that the lower the value of χ,
the lower the value of ET, but only when the condition of
matched velocities v1 ≈ v2 is fulfilled. Ultimately, χ = 0 corresponds
350
Results and discussion
Measurements. Experimental ICSs in arbitrary units were obtained
from the averaged REMPI signal intensities I and the relative
velocity vr of the CO and He beams as σ ∝ I/(vr〈Δt〉), where 〈Δt〉 is
the mean interaction time between the two beam pulses, which
takes full account of the density-to-flux transformation under our
working conditions17. For near-threshold collisions with very little
recoil energy as studied here, 〈Δt〉 depends on χ especially for low
values (Supplementary Fig. 1). This is due to the accumulation of
CO( j = 1) molecules produced with almost no recoil velocity,
which are therefore travelling in phase with the centre-of-mass
0.8
0.6
j=0
j=1
0.6
0.4
0.4
0.2
0.2
0.0
Intensity S(1) (×20)
temperatures. These rate coefficients, calculated by averaging the
product ICS × vr over a thermal distribution of relative velocities
vr , need to be accurately known for modelling the physics and
chemistry of the interstellar medium, in particular the cold cores
of dense molecular clouds where T falls to the 10 K range. It is
well established that in the interstellar medium, collisions
compete with radiative processes in altering the populations of molecular ro-vibrational levels. Estimates of the molecular abundances
from spectral line data require knowledge of collisional rate coefficients with H2 , by far the most abundant species, but also with
He, which follows immediately in column density.
Inelastic scattering cross-sections are very sensitive to the position in energy and the magnitude and width of resonance features.
Calculation of these features by quantum-mechanical methods
strongly depends on the quality of the multidimensional potential
energy surface (PES) used to describe the interaction. Validation
by comparison with experimental data is thus essential. However,
a temperature of T = 10 K in a medium at thermodynamic equilibrium corresponds to a mean kinetic energy of 〈ET〉 = 3/2kBT
(kB is the Boltzmann constant) of 125 J mol−1 or 10 cm−1.
Obtaining such low energies in single collision conditions (for
example, as provided by crossed-beam scattering experiments)
with sufficient resolution for resolving the resonance structures is
a challenge17. When two molecular species with well-defined velocities v1 and v2 collide at an angle χ, the relative translational
energy (collision energy) is given by
to merged-beam experiments where collision energies well below
the upper limit of the cold regime (T < 1 K, ET < 1 cm−1) can be
sampled18–20. Merged beams have been widely used in the past for
ion–molecule reactions, but recent experimental breakthroughs
have extended the technique to collisions between neutral species.
Such experiments have enabled the first observation of orbiting resonances in a chemical reaction occurring in the cold regime, for the
Penning ionization He(3S) + H2(D2 , HD) → He + H2+(D2+, HD+) + e−
(refs 18,19).
Although scattering resonances were observed some time ago in
low-energy elastic collisions21,22, the finding of such quantum effects
in inelastic collision experiments is only very recent, with first
reports obtained in conventional crossed beams for CO + H2
(ref. 4) and O2 + H2 (refs 5,6). It is interesting to note that previous
crossed-beam studies on CO + He were conducted at collision energies above 500 cm−1, and could only probe the repulsive wall of the
PES23. In the experiment described here we used a set-up with variable beam crossing angle, which allowed us to tune the collision
energy in the threshold region of the CO( j = 0 → 1) rotational transition at 3.85 cm−1. The CO molecules were detected by resonanceenhanced multiphoton ionization (REMPI) (Fig. 2)24,25. These
experiments, as well as the quantum scattering close-coupling calculations performed on two different PESs calculated (1) at the singles
and doubles coupled cluster level with perturbative triples and extrapolation to the complete basis set (CCSD(T)/CBS)26 and (2) with
symmetry-adapted perturbation theory (SAPT)27 are described in
the Methods.
Intensity S(0)
20
DOI: 10.1038/NCHEM.2204
0.0
400
450
500
Time of arrival (µs)
Figure 2 | Temporal profiles of the CO beam in the beam crossing region.
REMPI intensities (in arbitrary units, a.u.) for CO j = 0 and 1 rotational states
obtained with (E1Π, v = 0 ← X1Σ+g, v = 0) S(0) (blue circles) and S(1) (red
triangles) transitions. Vertical error bars represent statistical fluctuations at a
95% confidence interval after averaging over 100 laser shots. The profiles
are the result of efficient CO–Ne inelastic collisions that occur in the
supersonic expansion of the carrier gas immediately after the pulsed nozzle.
The scattering experiment is synchronized around 450 µs, in the coldest
part of the molecular beam, where the ratio of number densities of
CO( j = 0) versus CO( j = 1) is the highest.
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ICSexp (a.u.)
Partial waves (nm2)
Partial
waves (nm2)
IV
0.2
II
0.4
0.6
V
ET
0.0
(cm–1)
0.4
4.0 4.5 4.0 4.5 6.0
0.2
0.2
0.0
0.0
J=0
J=6
0.3
J=2
J=8
ICSth (nm2)
b
ICS
I
III
c
a
0.4
J=4
J = 10
0.2
0.1
[Wavefunction]2 (a.u.)
a
0.6
ARTICLES
DOI: 10.1038/NCHEM.2204
j, I
1, 5
1, 3
0, 4
8
6
4
2
0
5
10
15
20
25
30
R (bohr)
b
[Wavefunction]2 (a.u.)
NATURE CHEMISTRY
j, I
2, 3
1, 4
0, 5
10
5
0
5
10
15
20
25
30
R (bohr)
0.0
J=1
J=7
0.1
J=3
J=9
J=5
0.0
5
10
15
20
25
Relative translational energy (cm–1)
Figure 3 | Experimental and theoretical cross-sections for CO( j = 0) +
He → CO( j = 1) + He inelastic collisions. a, Experimental ICSs (open circles
and triangles) result from two sets of experiments (Supplementary Table 1)
and are corrected data, σ ∝ I/(vr 〈Δt〉). Vertical error bars represent
statistical fluctuations at a 95% confidence interval. To facilitate comparison
of experiment and theory, the total calculated ICSs (dashed curves) are
convoluted (solid curves) with the experimental resolution resulting from the
velocity spread and angular divergence of the beams: orange and green
curves were calculated using CCSD(T)/CBS and SAPT PESs, respectively. To
allow for comparison, data are normalized relative to the ICS calculated with
the SAPT PES. This results in a rescaling of the displayed CCSD(T)/CBS
ICSs by a factor of 0.975. The main peaks, labelled I, II, III, IV and V,
correspond to the resonances discussed in the text. b,c, Partial crosssections from quantum-mechanical calculations using the CCSD(T)/CBS
PES, for even and odd values of total angular momentum J.
velocity vector (Supplementary Fig. 2). It induces an angular spread
and a collision energy spread (Supplementary Fig. 1) that are
dependent on the beam intersection angle χ0. It also results in a
shift Δχ of the mean value of crossing angle χ towards values
higher than nominal: χ = χ 0 + Δχ (Supplementary Fig. 1). In other
words, (1) the REMPI signal intensity appears enhanced at
threshold and (2) the mean collision energy at angle χ0 is slightly
higher than the nominal one calculated using equation (1) with
χ = χ0. All details on the respective calculations of these quantities
are presented elsewhere17.
Cross-sections and resonances. Figure 3 compares the two sets of
experimental results obtained with two CO beams of slightly
different velocities (Supplementary Table 1) with those of the
close-coupling calculations performed with the CCSD(T)/CBS
and SAPT potentials. It can be seen in Fig. 3a that the ICSs
obtained with the two PESs are very similar and that the
experimental data are in very good agreement with the theoretical
ICSs curves convoluted over the collision energy spread. Many
strong resonances occur for collision energies in the vicinity of
the energetic threshold of the j = 1 channel. The five peaks
labelled I, II, III, IV and V, respectively, mainly result from
overlapped contributions of partial waves with total angular
Figure 4 | Wavefunctions squared, at the resonance energy, as a function
of radial distance. a, Wavefunctions computed at the maximum of peak I,
ET = 5.48 cm−1, corresponding to total angular momentum J = 4. The
dominant contribution in the well region corresponds to the open channel
with j = 1 and l = 5. Note that the amplitude of this contribution extends to
relatively long distances out of the well (by tunnelling through the
centrifugal barrier) and also contains a continuum contribution, which is
characteristic of a shape resonance. b, Wavefunctions computed at the
maximum of peak III, ET = 8.76 cm−1, corresponding to total angular
momentum of J = 5. The dominant contribution in the well region
corresponds to the closed channel with j = 2 and l = 3. Its amplitude is
restricted to shorter distances within the van der Waals well and it has no
continuum contribution, which is characteristic of a Feshbach resonance.
momentum J = 2 to 6 shown in Fig. 3b for even J values and in
Fig. 3c for odd values. Note that the total angular momentum J is
the sum of the orbital angular momentum l and the CO
rotational angular momentum j. Specifically, peak I mainly takes
its intensity from partial waves J = 4 and 6, with a contribution of
J = 3 in the falling edge, peak II from J = 2 and 4, peak III from
J = 3 and 5, peak IV from J = 4 and 6 and peak V from J = 5. The
collision energy spread (Supplementary Fig. 1), although good for
a crossed-beam scattering experiment does not allow for full
resolution of the peaks, but I, III and IV are still discernible in the
ICS and V can be distinguished as a shoulder. Peak II, which is a
feature of lower intensity, is smoothed in the experimental
bump of peak III.
The resonances due to the occurrence of quasi-bound states
ephemerally accessible during the collision were characterized by
generating the scattering wavefunctions at the maxima of the
partial wave contributions. These scattering wavefunctions calculated for specific values of J show, on resonance, large contributions
from a specific CO rotational state j and orbital angular momentum
l in the region of the van der Waals well. For example, at
ET = 5.48 cm−1, partial waves with J = 4 and 6 and l = 5 correlate
with j = 1, which is open at this energy (Ej=1 = 3.85 cm−1), and tunnelling through the centrifugal barrier gives rise to a shape resonance. Figure 4a gives a clear demonstration of the tunnelling. The
scattering wavefunction for the open channel with J = 4, l = 5 and
j = 1 is a plane wave at very large R and has large amplitude in
the van der Waals well, with a tail in the region of the centrifugal
barrier. At ET = 8.76 cm−1, partial waves with J = 3 and 5 and l = 3
correlate with j = 2 (Fig. 4b), which is an asymptotically closed
channel (Ej=2 = 11.54 cm−1). When the collision partners approach
the region of the van der Waals well around R = 8 bohr, CO ephemerally accesses the j = 2 state, which is energetically allowed only at
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Table 1 | Resonances in inelastic CO( j = 0) + He →
CO( j = 1) + He cross-sections calculated with the
CCSD(T)/CBS and SAPT potentials.
Energy (cm−1)
CCSD(T)/CBS
5.48
6.00
7.16
8.14
8.76
10.82
13.22
Energy (cm−1)
SAPT
5.30
5.65
6.82
7.96
8.45
10.57
13.04
J
Type of resonance
4, 6
3
2, 4
5
3, 5
4, 6
5
j = 1, l = 5, shape
j = 2, l = 1, Feshbach
j = 2, l = 2, Feshbach
j = 1, l = 6, shape
j = 2, l = 3, Feshbach
j = 2, l = 4, Feshbach
j = 2, l = 5, Feshbach + shape
because j = 2 is open
these distances and closed at larger R. As the collision partners move
off, the transient CO( j = 2) molecules relax back to the open j = 1
state. As shown in Fig. 4b, the j = 2 contribution to the scattering
wavefunction is confined to radial distances inside the well, which
is the signature of a Feshbach resonance. Other calculations (summarized in Table 1) establish the nature of all resonances, shape
or Feshbach, in the different partial wave contributions.
Figure 3 and Table 1 show that the resonances are systematically
lower in energy by 0.2–0.3 cm−1 in the SAPT potential than in the
CCSD(T)/CBS potential. This is related to a difference of 0.36
cm−1 between the dissociation energies D0 = 6.79 cm−1 (SAPT)
and D0 = 6.43 cm−1 (CCSD(T)/CBS). In turn, this is related to the
different binding energies De = 23.73 cm−1 and 22.34 cm−1, respectively, and perhaps to the fact that the two potentials differ in the
shape of the (very flat) van der Waals well. This implies that the
lowest bound level is lower by 0.36 cm−1 in the SAPT potential
than in the CCSD(T)/CBS potential. Also, the higher bound levels
calculated with the SAPT potential are lower by about the same
amount and, obviously, this holds for the resonances as well.
As mentioned above, the overall agreement between the experimental data and the quantum-mechanical calculations with both
potentials is very good, but the sharp rise of the ICS in the threshold
region is better characterized by the SAPT potential where the mismatch (inset of Fig. 3a) between experimental and theoretical
points does not exceed 0.2 cm−1 (instead of 0.4 cm−1 with the
CCSD(T)/CBS potential). However, the energy spread in this
region varies from 1 to 1.5 cm−1, and the absolute uncertainty on
the nominal collision energy is on the order of magnitude of the
aforementioned differences. Therefore, it cannot be concluded
that a particular potential is better than the other.
Rate coefficients. The excellent agreement between experimental
and theoretical results validates the SAPT and CCSD(T)/CBS
potentials, in particular in the most delicate parts to calculate
accurately: the long-range part and the van der Waals well. Either
PES can be used with confidence to calculate precise lowtemperature rate coefficients for rotational (de)excitation of CO by
He for inclusion in astrochemical models. These rate coefficients
are of particular importance (after those between CO and H2), as
CO is universally used to determine the temperature in various
regions of the interstellar medium including the coldest—dense
molecular clouds—through transitions in the millimetre
wavelength range. The rate coefficients computed with the
CCSD(T)/CBS and SAPT potentials are listed in Supplementary
Table 2. The two potentials yield only slightly different rate
coefficients. Our results from the SAPT potential agree nicely
with the results calculated previously3 with the same potential for
the lower j-values of CO, but they differ from those results by up
to 10% for higher j, especially at low temperatures. However,
Balakrishnan, one of the authors of ref. 3, repeated their
calculations, and his present results agree with ours.
352
DOI: 10.1038/NCHEM.2204
Conclusions
Our results highlight the quantum-mechanical nature of the
CO( j = 0) + He → CO( j = 1) + He inelastic collision process in
the low-energy region. Despite the partial wave averaging and the
collision energy spread, resonance features are still visible in the
experimental ICSs and are completely characterized as shape and
Feshbach resonances.
Methods
In the experiments, CO and He beams with low and matched velocities and high
velocity resolution (Supplementary Table 1) were obtained using cryogenically
cooled Even–Lavie pulsed valves28 and collided at a beam intersection angle that
could be varied continuously from 90° to 12.5° (ref. 17). The CO molecules were
detected in the beam crossing region by (2 + 1) REMPI time-of-flight mass
spectrometry (TOF MS) using two-photon (E1Π, v = 0 ← X1Σ+g , v = 0) transitions
around 215.217 nm (refs 24,25). Due to efficient cooling by collisions with Ne
carrier gas in the supersonic expansion only very weak S(1), R(1) and Q(1) lines
relative to j = 1 were detected in the vicinity of the intense S(0) transition
characterizing j = 0 (ref. 4). Obviously, cooling is at its best in the centre of the
gas pulse (Fig. 2) where the population ratio N( j = 1)/N( j = 0) is estimated to be
<0.01 and the rotational temperature T ≈ 0.9 K. The ICSs were obtained with
the probe laser tuned to the S(1) line, the wavelength of which was continuously
checked by a high-precision wavemeter. The background due to the residual
population of j = 1 was subtracted by pulsing the laser and the CO beam at 10 Hz
while triggering the He beam at 5 Hz. Signal fluctuations were reduced by
accumulating short-duration scans (typically 400 s) to avoid long-term drifts. In
Fig. 3, each point of the lower (respectively higher) energy set corresponds to 40
consecutive scans of the beam intersection angle acquired between 22.5°
(respectively 30°) and 12.5° with –0.5° decrement and 150 (respectively 100)
laser shots per angle.
Quantum-mechanical close-coupling calculations were performed with an
in-house scattering program for molecule–molecule collisions that had been used
previously for He–NH3 (ref. 13). As stressed in the introduction, an accurate PES is
an essential ingredient of the scattering calculations. Five three-dimensional PESs
were published in 2005 for CO–He, four of which were obtained at the CCSD(T)
level with different basis sets26. The fifth PES, which should be the most accurate, is
an approximate CCSD(T)/CBS surface. The two-dimensional potential obtained by
vibrationally averaging the latter PES over the v = 0 ground state of CO is used for
the present scattering calculations. This interaction potential depends on the two
coordinates R and γ, where R is the length of the vector R that points from the CO
centre-of-mass to the He atom and γ is the angle between R and the C–O bond axis.
In addition, we used another two-dimensional CO–He PES that was obtained by
CO(v = 0) vibrational averaging of a three-dimensional SAPT potential27. Both
interaction potentials were expanded in Legendre polynomials PL(γ) with the
expansion coefficients depending on R. The maximum L-value used was 10, and it
was checked that a larger value yields practically the same results. In the scattering
calculations the close-coupling equations were solved with the renormalized
Numerov propagator, with R ranging from 4 to 50 bohr in 217 steps. All rotational
states of CO up to j = 20 were included, and we took into account all partial wave
contributions up to a total angular momentum of J = 20 to reach convergence.
Integral cross-sections and the individual partial wave contributions were computed
for collision energies ranging from 0 to 50 cm−1 with increments of 0.02 cm−1.
The corresponding values used in the calculation of the rate coefficients are given in the
caption of Supplementary Table 2. Finally, we tested the convergence of our
calculations extremely carefully and checked our results by recomputing them with
a different scattering program that uses the full three-dimensional SAPT potential,
instead of its two-dimensional version obtained by averaging over the CO(v = 0) state.
Received 22 October 2014; accepted 12 February 2015;
published online 24 March 2015
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Acknowledgements
This work extends the objectives of the ANR-12-BS05-0011-02 contract with the Agence
Nationale de la Recherche and contract no. 2007.1221 with the Conseil Régional
d’Aquitaine, for which financial support is acknowledged. The authors acknowledge
support from Partenariat Hubert Curien van Gogh (contract 2013-28484TH). The authors
thank L. Song for help with scattering calculations on the full three-dimensional SAPT
potential and N. Balakrishnan for checking the results of ref. 3.
Author contributions
A.B., C.N. and M.C. carried out the experimental measurements and data analysis. J.O. and
A.v.d.A. performed the theoretical calculations. All authors discussed the results and
contributed to the manuscript.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints. Correspondence and
requests for materials should be addressed to M.C. and A.v.d.A.
Competing financial interests
The authors declare no competing financial interests.
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